Kinematic Hopf algebra for amplitudes and form factors

Abstract

We propose a kinematic algebra for the Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes and form factors in Yang-Mills theory coupled with biadjoint scalars. The algebraic generators of the algebra contain two parts: the first part is simply the flavor factor of the biadjoint scalars, and the second part that maps to nontrivial kinematic structures of the BCJ numerators obeys extended quasishuffle fusion products. The underlying kinematic algebra allows us to present closed forms for the BCJ numerators with any number of gluons and two or more scalars for both on-shell amplitudes and form factors that involve an off-shell operator. The BCJ numerators constructed in this way are manifestly gauge invariant and obey many novel relations that are inherited from the kinematic algebra.